Newton line



In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.

Properties
The line segments $E, K, F$ and $\overline{GH}$ that connect the midpoints of opposite sides (the bimedians) of a convex quadrilateral intersect in a point that lies on the Newton line. This point $\overline{IJ}$ bisects the line segment $K$ that connects the diagonal midpoints.

By Anne's theorem and its converse, any interior point P on the Newton line of a quadrilateral $\overline{EF}$ has the property that
 * $$[\triangle ABP] + [\triangle CDP] = [\triangle ADP] + [\triangle BCP],$$

where $[△ABP]$ denotes the area of triangle $△ABP$.

If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line.